A centrifugal compressor element has a high efficiency when its specific speed is situated close to the optimal value. The specific speed Ns is defined as:
  Ns  =            C      ′        ·                  N        ·                              Q            vol                                      DH                  0.          ⁢                                          ⁢          75                    whereby:    N=the rotational speed of the blade wheel,    Qvol=the volumetric flow on the inlet,    C′=a constant which is amongst others different as a function of the units used,    DH=the adiabatic head of the compressor:
  DH  =      cp    ·    T    ·          (                        π                                    k              -              1                        k                          -        1            )      whereby:    π=the pressure ratio,    T=the inlet temperature,    cp=the specific heat of the gas at a constant pressure,    k=the ratio of the specific heat of the gas at the constant pressure and the specific heat of the gas at a constant volume.
In order to obtain a good efficiency, and thus a low specific consumption or energy consumption per quantity of compressed air, it is necessary to select the parameters in the design of a compressor element such that Ns is situated close to the optimum.
In fact, the equation for Ns indicates that for designs having the same flow, the rotational speed has to rise for a higher pressure ratio, and for designs with a constant pressure ratio, the rotational speed has to rise for a smaller flow.
Centrifugal compressors are known whereby the shafts of the compressor elements are driven directly by electric motors at a high speed of rotation.
Such centrifugal compressors require less stages to obtain a high pressure ratio than the conventional centrifugal compressors which are driven directly by high-speed motors at a low speed.
High-speed motors are characterised by a characteristic value M=P.N2 which is larger than or equal to 0,1.1012, whereby P is the engine power expressed in kW and N is the rotational speed expressed in rotations per minute.
The fast drive allows for a higher pressure ratio per stage. Less stages means less loss.
Such centrifugal compressors avoid the use of a gearbox as in conventional centrifugal compressors with a drive via a gearbox which implies a great deal of losses, requires oiling and occupies much space.
Moreover, a high-speed motor is much smaller than a conventional, slow electric motor.
The high-speed motor is equipped with adjusted bearings for these high rotational speeds. When air bearings or magnetic bearings are used, no oil is required, and the compressor is entirely oil-free, which offers an additional advantage in relation to compressors with bearings requiring oil lubrication.
The problem resides in the restriction of the power and the rotational speed of the high-speed motor, and the needs for a centrifugal compressor for high pressure.
Electric high-speed motors are characterised by a small volume and consequently a high energy density. Given the small dimensions, the cooling causes a specific problem.
The ratio of the applied power P and the dischargeable power (h.A) is the dimensionless value M′=P/(h.A). Hereby is A the reference heat-exchanging surface, and h is the effective heat transfer coefficient between the hot motor and the colder environment, possibly via a cooling system with heat exchanger.
The surface is proportional to the square of the specific length of the motor, namely the radius of the rotor R. Also the characteristic value M′ can be represented as:
      M    ′    =      P          h      ·              R        2            
The radius of the rotor also is the relation of V to N, whereby N is the rotational speed of the motor and V is the tip speed of the rotor. Thus, M′ can be represented as:
      M    ′    =            P      ·              N        2                    h      ·              V        2            
For a given type of heat exchange, h is a constant, and for a given material, V is restricted as a result of centrifugal tensions.
Consequently, the characteristic value M=P.N2 is a value which indicates the level of difficulty of the design and the construction of the electric motor. The higher the value M, the more difficult it is to cool the motor. A high value M requires more efficiency (so that less losses have to be discharged), a better heat transfer coefficient and a higher strength of material.
In practice, this implies that a motor having a higher characteristic value M requires a more expensive design, and that the development will take longer than for a motor having a lower characteristic value M.
For a turbocompressor, the power required is equal to:
  P  =                    Q        ·        DH            η        =                  ρ        ·                  Q          vol                ·        cp        ·        T        ·                  (                                    π                                                k                  -                  1                                k                                      -            1                    )                    η      whereby:    θ=the adiabatic efficiency of the compressor,    ρ=the density of the gas,    Q=the mass flow.
The number of revolutions N is selected as a function of a good specific rotational speed Ns
  N  =            Ns      ·              DH                  0          .                                          ⁢          75                                    C        ′            ·                        Q          vol                    from which appears the following:
  M  =            P      ·              N        2              =                                                      Ns              2                        ·                          cp                              2                .                                                                  ⁢                5                                                                        C              ′2                        ·            η                          ·        ρ        ·                              [                          T              ·                              (                                                      π                                                                  k                        -                        1                                            k                                                        -                  1                                )                                      ]                    2.5                    =              C        ·        ρ        ·                              [                          T              ·                              (                                                      π                                                                  k                        -                        1                                            k                                                        -                  1                                )                                      ]                                2.            ⁢                                                  ⁢            5                              C is hereby a constant. This equation indicates that an electric motor for a centrifugal compressor which is driven directly is more difficult to realise for a higher pressure ratio (π) and for a high-pressure stage, this is with a higher density at the inlet.
It is clear from this argumentation that a compression to high pressures in a single stage is extremely difficult to realise with a single drive.
That is why a solution must be found to nevertheless keep the characteristic value M low.
An obvious solution is to carry out the compression in more than one stage, thereby using more than one motor, for example one motor for the low-pressure stage and one motor for the high-pressure stage.
However, from the last equation it is clear that the higher pressure for the high-pressure stage is coupled with a much higher characteristic value M. This is difficult to realise.
Consequently, the designer has to be content with a lower Ns and hence less efficiency.
A restricted improvement can be obtained by providing for an optimal distribution of the pressure ratios of the low- and high-pressure stages, namely by setting the pressure ratio in the first stages higher than the pressure ratios of the last stages.
However, said improvement is restricted, since for pressure ratio's which are larger than three, the Mach value losses (shock losses) strongly increase.